A sphere (from Greek σφαῖρα, sphaîra)^[1] is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. Formally, a sphere is the set of points that are all at the same distance $r$ from a given point in three-dimensional space.^[2] That given point is the center of the sphere, and $r$ is the sphere's radius. The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians.

The sphere is a fundamental object in many fields of mathematics. Spheres and nearly-spherical shapes also appear in nature and industry. Bubbles such as soap bubbles take a spherical shape in equilibrium. The Earth is often approximated as a sphere in geography, and the celestial sphere is an important concept in astronomy. Manufactured items including pressure vessels and most curved mirrors and lenses are based on spheres. Spheres roll smoothly in any direction, so most balls used in sports and toys are spherical, as are ball bearings.

Basic terminology[edit]

As mentioned earlier $r$ is the sphere's radius; any line from the center to a point on the sphere is also called a radius.^[3]

If a radius is extended through the center to the opposite side of the sphere, it creates a diameter. Like the radius, the length of a diameter is also called the diameter, and denoted $d$ . Diameters are the longest line segments that can be drawn between two points on the sphere: their length is twice the radius, $d = 2 r$ . Two points on the sphere connected by a diameter are antipodal points of each other.^[3]

A unit sphere is a sphere with unit radius ( $r = 1$ ). For convenience, spheres are often taken to have their center at the origin of the coordinate system, and spheres in this article have their center at the origin unless a center is mentioned.

A great circle on the sphere has the same center and radius as the sphere, and divides it into two equal hemispheres.

Although the figure of Earth is not perfectly spherical, terms borrowed from geography are convenient to apply to the sphere. A particular line passing through its center defines an axis (as in Earth's axis of rotation). The sphere-axis intersection defines two antipodal poles (north pole and south pole). The great circle equidistant to the poles is called the equator. Great circles through the poles are called lines of longitude or meridians. Small circles on the sphere that are parallel to the equator are circles of latitude (or parallels). In geometry unrelated to astronomical bodies, geocentric terminology should be used only for illustration and noted as such, unless there is no chance of misunderstanding.^[3]

Mathematicians consider a sphere to be a two-dimensional closed surface embedded in three-dimensional Euclidean space. They draw a distinction between a sphere and a ball, which is a three-dimensional manifold with boundary that includes the volume contained by the sphere. An open ball excludes the sphere itself, while a closed ball includes the sphere: a closed ball is the union of the open ball and the sphere, and a sphere is the boundary of a (closed or open) ball. The distinction between ball and sphere has not always been maintained and especially older mathematical references talk about a sphere as a solid. The distinction between "circle" and "disk" in the plane is similar.

Small spheres or balls are sometimes called spherules, e.g. in Martian spherules.

Enclosed volume[edit]

In three dimensions, the volume inside a sphere (that is, the volume of a ball, but classically referred to as the volume of a sphere) is

V={\frac {4}{3}}\pi r^{3}={\frac {\pi }{6}}\ d^{3}\approx 0.5236\cdot d^{3}

where $r$ is the radius and $d$ is the diameter of the sphere. Archimedes first derived this formula by showing that the volume inside a sphere is twice the volume between the sphere and the circumscribed cylinder of that sphere (having the height and diameter equal to the diameter of the sphere).^[6] This may be proved by inscribing a cone upside down into semi-sphere, noting that the area of a cross section of the cone plus the area of a cross section of the sphere is the same as the area of the cross section of the circumscribing cylinder, and applying Cavalieri's principle.^[7] This formula can also be derived using integral calculus, i.e. disk integration to sum the volumes of an infinite number of circular disks of infinitesimally small thickness stacked side by side and centered along the $x$ -axis from $x = - r$ to $x = r$ , assuming the sphere of radius $r$ is centered at the origin.

Other geometric properties[edit]

A sphere can be constructed as the surface formed by rotating a circle one half revolution about any of its diameters; this is very similar to the traditional definition of a sphere as given in Euclid's Elements. Since a circle is a special type of ellipse, a sphere is a special type of ellipsoid of revolution. Replacing the circle with an ellipse rotated about its major axis, the shape becomes a prolate spheroid; rotated about the minor axis, an oblate spheroid.^[12]

A sphere is uniquely determined by four points that are not coplanar. More generally, a sphere is uniquely determined by four conditions such as passing through a point, being tangent to a plane, etc.^[13] This property is analogous to the property that three non-collinear points determine a unique circle in a plane.

Consequently, a sphere is uniquely determined by (that is, passes through) a circle and a point not in the plane of that circle.

By examining the common solutions of the equations of two spheres, it can be seen that two spheres intersect in a circle and the plane containing that circle is called the radical plane of the intersecting spheres.^[14] Although the radical plane is a real plane, the circle may be imaginary (the spheres have no real point in common) or consist of a single point (the spheres are tangent at that point).^[15]

The angle between two spheres at a real point of intersection is the dihedral angle determined by the tangent planes to the spheres at that point. Two spheres intersect at the same angle at all points of their circle of intersection.^[16] They intersect at right angles (are orthogonal) if and only if the square of the distance between their centers is equal to the sum of the squares of their radii.^[4]

In geometry, the area enclosed by a circle of radius $r$ is $π r 2$ . Here the Greek letter $π$ represents the constant ratio of the circumference of any circle to its diameter, approximately equal to 3.14159.

One method of deriving this formula, which originated with Archimedes, involves viewing the circle as the limit of a sequence of regular polygons with an increasing number of sides. The area of a regular polygon is half its perimeter multiplied by the distance from its center to its sides, and because the sequence tends to a circle, the corresponding formula–that the area is half the circumference times the radius–namely, $A = 1 / 2 \times 2π r \times r$ , holds for a circle.

Astronomical coordinate systems

Sphere
Sphere
	A perspective projection of a sphere
Type	Smooth surface; Algebraic surface
Euler char.	2
Symmetry group	O(3)
Surface area	4πr2
Volume	4/3πr3

From Wikipedia, the free encyclopedia

(Redirected from Celestial coordinate system)

In astronomy, coordinate systems are used for specifying positions of celestial objects (satellites, planets, stars, galaxies, etc.) relative to a given reference frame, based on physical reference points available to a situated observer (e.g. the true horizon and north to an observer on Earth's surface).^[1] Coordinate systems in astronomy can specify an object's position in three-dimensional space or plot merely its direction on a celestial sphere, if the object's distance is unknown or trivial.

Spherical coordinates, projected on the celestial sphere, are analogous to the geographic coordinate system used on the surface of Earth. These differ in their choice of fundamental plane, which divides the celestial sphere into two equal hemispheres along a great circle. Rectangular coordinates, in appropriate units, have the same fundamental ( $x, y$ ) plane and primary ( $x$ -axis) direction, such as an axis of rotation. Each coordinate system is named after its choice of fundamental plane.

Coordinate systems[edit]

The following table lists the common coordinate systems in use by the astronomical community. The fundamental plane divides the celestial sphere into two equal hemispheres and defines the baseline for the latitudinal coordinates, similar to the equator in the geographic coordinate system. The poles are located at ±90° from the fundamental plane. The primary direction is the starting point of the longitudinal coordinates. The origin is the zero distance point, the "center of the celestial sphere", although the definition of celestial sphere is ambiguous about the definition of its center point.

Star trail

From Wikipedia, the free encyclopedia

A star trail is a type of photograph that uses long exposure times to capture diurnal circles, the apparent motion of stars in the night sky due to Earth's rotation. A star-trail photograph shows individual stars as streaks across the image, with longer exposures yielding longer arcs. The term is used for similar photos captured elsewhere, such as on board the International Space Station and on Mars.^[2]^[3]

Typical shutter speeds for a star trail range from 15 minutes to several hours, requiring a "Bulb" setting on the camera to open the shutter for a period longer than usual. However, a more practiced technique is to blend a number of frames together to create the final star trail image.^[4]

Star trails have been used by professional astronomers to measure the quality of observing locations for major telescopes.

Capture[edit]

Video

Star trail photographs are captured by placing a camera on a tripod, pointing the lens toward the night sky, and allowing the shutter to stay open for a long period of time.^[5] Star trails are considered relatively easy for amateur astrophotographers to create.^[6] Photographers generally make these images by using a DSLR or Mirrorless camera with its lens focus set to infinity. A cable release or intervalometer allows the photographer to hold the shutter open for the desired amount of time. Typical exposure times range from 15 minutes to many hours long, depending on the desired length of the star trail arcs for the image.^[7] Even though star trail pictures are created under low-light conditions, long exposure times allow fast films, such as ISO 200 and ISO 400.^[6] Wide-apertures, such as f/5.6 and f/4, are recommended for star trails.^[5]^[8]

Because exposure times for star trail photographs can be several hours long, camera batteries can be easily depleted. Mechanical cameras that do not require a battery to open and close the shutter have an advantage over more modern film and digital cameras that rely on battery power. On these cameras, the Bulb, or B, exposure setting keeps the shutter open.^[9] Another problem that digital cameras encounter is an increase in electronic noise with increasing exposure time.^[5] However, this can be avoided through the use of shorter exposure times that are then stacked in post production software. This avoids possible heat build up or digital noise caused from a single long exposure.

American astronaut Don Pettit recorded star trails with a digital camera from the International Space Station in Earth orbit between April and June, 2012. Pettit described his technique as follows: "My star trail images are made by taking a time exposure of about 10 to 15 minutes. However, with modern digital cameras, 30 seconds is about the longest exposure possible, due to electronic detector noise effectively snowing out the image. To achieve the longer exposures I do what many amateur astronomers do. I take multiple 30-second exposures, then 'stack' them using imaging software, thus producing the longer exposure."^[10]^[2]

Star trail images have also been taken on Mars.^[11] The Spirit rover produced them while looking for meteors.^[11] Since the camera was limited to 60 second exposures the trails appear as dashed lines.^[11]

Earth's rotation[edit]

Star trail photographs are possible because of the rotation of Earth about its axis. The apparent motion of the stars is recorded as mostly curved streaks on the film or detector.^[5] For observers in the Northern Hemisphere, aiming the camera northward creates an image with concentric circular arcs centered on the north celestial pole (very near Polaris).^[6] For those in the Southern Hemisphere, this same effect is achieved by aiming the camera southward. In this case, the arc streaks are centered on the south celestial pole (near Sigma Octantis). Aiming the camera eastward or westward shows straight streaks on the celestial equator, which is tilted at angle with respect to the horizon. The angular measure of this tilt depends on the photographer's latitude^[5] ( $L$ ), and is equal to $90° - L$ .

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